I'll have more time to write and provide a more thorough answer later, but I think the most straightforward proof (which I agree is hard to find) comes via sheaf theory: On the one hand, there is a sheaf of locally finite singular chains whose hypercohomology is your $H^{lf}$. I work out the details in the setting of intersection homology on pseudomanifolds in Section 3 of the following paper, but the ordinary (non-intersection) lf homology on locally compact (paracompact?) spaces is essentially a special case via basically the same arguments: "Singular chain intersection homology for traditional and super-perversities" Transactions of the American Mathematical Society 359 (2007), 1977-2019
On the other hand, you can find a good treatment of the dualizing complex in Borel's book on intersection homology, section V.7. The idea then is to show that these two sheaves are quasi-isomorphic. This is the part I'll have to think about and add latter, though I'll note that it's not so hard for manifolds: in that case, it's not so hard to show by hand that both have to be quasi-isomorphic to the orientation sheaf (shifted appropriately). That's shown in Borel for the dualizing complex, and follows for the sheaf of locally finite singular chains via the local homology computation on the manifold.
Sorry I don't have time to write more now, but hopefully that's a start on the ideas.